import Mathlib.MeasureTheory.Order.Lattice
import Mathlib.Order.CompletePartialOrder
import Mathlib.Probability.Martingale.BorelCantelli
import Mathlib.Probability.Kernel.IonescuTulcea.Traj
import Mathlib.Probability.Process.FiniteDimensionalLaws
import Mathlib.Probability.HasCondDistrib
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.Probability.Independence.Basic
import Mathlib.Probability.Independence.Conditional
import Mathlib.MeasureTheory.Measure.SubFinite
import Mathlib.Probability.Kernel.RadonNikodym
import Mathlib.Analysis.Normed.Ring.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

/-! # Standalone extraction for `Bandits.regret_nonneg`
Definitions are copied verbatim; theorem proofs are replaced by `sorry`.
Auto-generated by Referee. -/

set_option quotPrecheck false

-- Namespace stubs (so later `open`s resolve).
namespace Finset
end Finset
namespace MeasureTheory
end MeasureTheory
namespace ProbabilityTheory
end ProbabilityTheory
namespace Learning
end Learning
namespace ENNReal
end ENNReal
namespace Bandits
end Bandits

-- ═══ ForMathlib.MeasureTheory.Order.Lattice ═══
section
open Finset
variable {α δ : Type*} [MeasurableSpace δ] [SemilatticeInf α] {m : MeasurableSpace α} [MeasurableInf₂ α]
attribute [to_dual existing] MeasurableInf₂
end

-- ═══ Online.Bandit.Regret ═══
section
open MeasureTheory ProbabilityTheory Filter Real Finset Learning
open scoped ENNReal NNReal
namespace Bandits
variable {𝓐 Ω : Type*} [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐} {mΩ : MeasurableSpace Ω} {ν : Kernel 𝓐 ℝ} {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ} {ω : Ω} {m n t : ℕ} {a : 𝓐}

/-- Regret of a sequence of pulls `k : ℕ → 𝓐` at time `t` for the reward kernel `ν ; Kernel 𝓐 ℝ`. -/
noncomputable
def regret (ν : Kernel 𝓐 ℝ) (A : ℕ → Ω → 𝓐) (t : ℕ) (ω : Ω) : ℝ :=
  t * (⨆ a, (ν a)[id]) - ∑ s ∈ range t, (ν (A s ω))[id]

omit [DecidableEq 𝓐] in
lemma regret_nonneg [Finite 𝓐] : 0 ≤ regret ν A t ω := sorry

end Bandits
end